C2^4.413C2^3 - GroupNames

G = C₂⁴.₄₁₃C₂³ order 128 = 2⁷

253^rd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂⁴.₄₁₃C₂³, C₂³.₆₁₄C₂⁴, C₂².₃₈₈2₊¹⁺⁴, C₄⋊C₄⋊₁₇D₄, C₂³⋊₂D₄⋊₄₃C₂, C₂.₃₁(Q₈⋊₆D₄), C₂³.₈₃(C₄○D₄), C₂.₁₁₉(D₄⋊₅D₄), C₂³.₁₁D₄⋊₉₅C₂, C₂³.₁₀D₄⋊₉₆C₂, C₂³.₂₃D₄⋊₉₉C₂, (C₂²×C₄).₁₉₂C₂³, (C₂×C₄²).₆₆₅C₂², (C₂³×C₄).₄₆₉C₂², C₂³.₈Q₈⋊₁₁₅C₂, C₂².₄₂₃(C₂²×D₄), C₂⁴.₃C₂²⋊₈₈C₂, (C₂²×D₄).₂₄₈C₂², C₂⁴.C₂²⋊₁₄₁C₂, C₂.₇₂(C₂².₃₂C₂⁴), C₂.₆₃(C₂².₂₉C₂⁴), C₂³.₆₃C₂³⋊₁₄₁C₂, C₂.₂₀(C₂².₅₄C₂⁴), C₂.C₄².₃₂₀C₂², C₂.₈₈(C₂².₄₇C₂⁴), (C₂×C₄⋊D₄)⋊₃₈C₂, (C₂×C₄).₁₁₂(C₂×D₄), (C₂×C₄²⋊₂C₂)⋊₂₂C₂, (C₂×C₄).₂₀₁(C₄○D₄), (C₂×C₄⋊C₄).₄₂₇C₂², C₂².₄₇₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₇₉C₂², SmallGroup(128,1446)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂³ — C₂⁴.₄₁₃C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.₃C₂² — C₂⁴.₄₁₃C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₁₃C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₁₃C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₁₃C₂³

Generators and relations for C₂⁴.₄₁₃C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=g²=a, f²=ba=ab, ac=ca, ede=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 692 in 299 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂×C₄⋊D₄, C₂×C₄²⋊₂C₂, C₂⁴.₄₁₃C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₂₉C₂⁴, C₂².₃₂C₂⁴, D₄⋊₅D₄, Q₈⋊₆D₄, C₂².₄₇C₂⁴, C₂².₅₄C₂⁴, C₂⁴.₄₁₃C₂³

Smallest permutation representation of C₂⁴.₄₁₃C₂³
►On 64 points

Generators in S₆₄

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)

(1 25)(2 26)(3 27)(4 28)(5 19)(6 20)(7 17)(8 18)(9 15)(10 16)(11 13)(12 14)(21 62)(22 63)(23 64)(24 61)(29 35)(30 36)(31 33)(32 34)(37 51)(38 52)(39 49)(40 50)(41 59)(42 60)(43 57)(44 58)(45 56)(46 53)(47 54)(48 55)

(1 31)(2 32)(3 29)(4 30)(5 53)(6 54)(7 55)(8 56)(9 37)(10 38)(11 39)(12 40)(13 49)(14 50)(15 51)(16 52)(17 48)(18 45)(19 46)(20 47)(21 42)(22 43)(23 44)(24 41)(25 33)(26 34)(27 35)(28 36)(57 63)(58 64)(59 61)(60 62)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)

(1 59)(2 58)(3 57)(4 60)(5 52)(6 51)(7 50)(8 49)(9 47)(10 46)(11 45)(12 48)(13 56)(14 55)(15 54)(16 53)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 41)(26 44)(27 43)(28 42)(29 63)(30 62)(31 61)(32 64)

(1 28 27 2)(3 26 25 4)(5 45 17 54)(6 53 18 48)(7 47 19 56)(8 55 20 46)(9 10 13 14)(11 12 15 16)(21 43 64 59)(22 58 61 42)(23 41 62 57)(24 60 63 44)(29 34 33 30)(31 36 35 32)(37 38 49 50)(39 40 51 52)

(1 9 3 11)(2 14 4 16)(5 62 7 64)(6 24 8 22)(10 26 12 28)(13 25 15 27)(17 23 19 21)(18 63 20 61)(29 39 31 37)(30 52 32 50)(33 51 35 49)(34 40 36 38)(41 56 43 54)(42 48 44 46)(45 57 47 59)(53 60 55 58)

G:=sub<Sym(64)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,25)(2,26)(3,27)(4,28)(5,19)(6,20)(7,17)(8,18)(9,15)(10,16)(11,13)(12,14)(21,62)(22,63)(23,64)(24,61)(29,35)(30,36)(31,33)(32,34)(37,51)(38,52)(39,49)(40,50)(41,59)(42,60)(43,57)(44,58)(45,56)(46,53)(47,54)(48,55), (1,31)(2,32)(3,29)(4,30)(5,53)(6,54)(7,55)(8,56)(9,37)(10,38)(11,39)(12,40)(13,49)(14,50)(15,51)(16,52)(17,48)(18,45)(19,46)(20,47)(21,42)(22,43)(23,44)(24,41)(25,33)(26,34)(27,35)(28,36)(57,63)(58,64)(59,61)(60,62), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,59)(2,58)(3,57)(4,60)(5,52)(6,51)(7,50)(8,49)(9,47)(10,46)(11,45)(12,48)(13,56)(14,55)(15,54)(16,53)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,41)(26,44)(27,43)(28,42)(29,63)(30,62)(31,61)(32,64), (1,28,27,2)(3,26,25,4)(5,45,17,54)(6,53,18,48)(7,47,19,56)(8,55,20,46)(9,10,13,14)(11,12,15,16)(21,43,64,59)(22,58,61,42)(23,41,62,57)(24,60,63,44)(29,34,33,30)(31,36,35,32)(37,38,49,50)(39,40,51,52), (1,9,3,11)(2,14,4,16)(5,62,7,64)(6,24,8,22)(10,26,12,28)(13,25,15,27)(17,23,19,21)(18,63,20,61)(29,39,31,37)(30,52,32,50)(33,51,35,49)(34,40,36,38)(41,56,43,54)(42,48,44,46)(45,57,47,59)(53,60,55,58)>;

G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,25)(2,26)(3,27)(4,28)(5,19)(6,20)(7,17)(8,18)(9,15)(10,16)(11,13)(12,14)(21,62)(22,63)(23,64)(24,61)(29,35)(30,36)(31,33)(32,34)(37,51)(38,52)(39,49)(40,50)(41,59)(42,60)(43,57)(44,58)(45,56)(46,53)(47,54)(48,55), (1,31)(2,32)(3,29)(4,30)(5,53)(6,54)(7,55)(8,56)(9,37)(10,38)(11,39)(12,40)(13,49)(14,50)(15,51)(16,52)(17,48)(18,45)(19,46)(20,47)(21,42)(22,43)(23,44)(24,41)(25,33)(26,34)(27,35)(28,36)(57,63)(58,64)(59,61)(60,62), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,59)(2,58)(3,57)(4,60)(5,52)(6,51)(7,50)(8,49)(9,47)(10,46)(11,45)(12,48)(13,56)(14,55)(15,54)(16,53)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,41)(26,44)(27,43)(28,42)(29,63)(30,62)(31,61)(32,64), (1,28,27,2)(3,26,25,4)(5,45,17,54)(6,53,18,48)(7,47,19,56)(8,55,20,46)(9,10,13,14)(11,12,15,16)(21,43,64,59)(22,58,61,42)(23,41,62,57)(24,60,63,44)(29,34,33,30)(31,36,35,32)(37,38,49,50)(39,40,51,52), (1,9,3,11)(2,14,4,16)(5,62,7,64)(6,24,8,22)(10,26,12,28)(13,25,15,27)(17,23,19,21)(18,63,20,61)(29,39,31,37)(30,52,32,50)(33,51,35,49)(34,40,36,38)(41,56,43,54)(42,48,44,46)(45,57,47,59)(53,60,55,58) );

G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,25),(2,26),(3,27),(4,28),(5,19),(6,20),(7,17),(8,18),(9,15),(10,16),(11,13),(12,14),(21,62),(22,63),(23,64),(24,61),(29,35),(30,36),(31,33),(32,34),(37,51),(38,52),(39,49),(40,50),(41,59),(42,60),(43,57),(44,58),(45,56),(46,53),(47,54),(48,55)], [(1,31),(2,32),(3,29),(4,30),(5,53),(6,54),(7,55),(8,56),(9,37),(10,38),(11,39),(12,40),(13,49),(14,50),(15,51),(16,52),(17,48),(18,45),(19,46),(20,47),(21,42),(22,43),(23,44),(24,41),(25,33),(26,34),(27,35),(28,36),(57,63),(58,64),(59,61),(60,62)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,59),(2,58),(3,57),(4,60),(5,52),(6,51),(7,50),(8,49),(9,47),(10,46),(11,45),(12,48),(13,56),(14,55),(15,54),(16,53),(17,40),(18,39),(19,38),(20,37),(21,36),(22,35),(23,34),(24,33),(25,41),(26,44),(27,43),(28,42),(29,63),(30,62),(31,61),(32,64)], [(1,28,27,2),(3,26,25,4),(5,45,17,54),(6,53,18,48),(7,47,19,56),(8,55,20,46),(9,10,13,14),(11,12,15,16),(21,43,64,59),(22,58,61,42),(23,41,62,57),(24,60,63,44),(29,34,33,30),(31,36,35,32),(37,38,49,50),(39,40,51,52)], [(1,9,3,11),(2,14,4,16),(5,62,7,64),(6,24,8,22),(10,26,12,28),(13,25,15,27),(17,23,19,21),(18,63,20,61),(29,39,31,37),(30,52,32,50),(33,51,35,49),(34,40,36,38),(41,56,43,54),(42,48,44,46),(45,57,47,59),(53,60,55,58)]])

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	4A	···	4P	4Q	4R	4S
order	1	2	···	2	2	2	2	2	2	4	···	4	4	4	4
size	1	1	···	1	4	4	8	8	8	4	···	4	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₄₁₃C₂³

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂×C₄⋊D₄

C₂×C₄²⋊₂C₂

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₁₃C₂³ ►in GL₆(𝔽₅)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

2	0	0	0	0	0
0	3	0	0	0	0
0	0	0	2	0	0
0	0	3	0	0	0
0	0	0	0	4	0
0	0	0	0	0	4

0	4	0	0	0	0
4	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	3
0	0	0	0	0	4

2	0	0	0	0	0
0	2	0	0	0	0
0	0	0	2	0	0
0	0	2	0	0	0
0	0	0	0	4	0
0	0	0	0	4	1

0	4	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,4,4,0,0,0,0,0,1],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂⁴.₄₁₃C₂³ in GAP, Magma, Sage, TeX

C_2^4._{413}C_2^3

% in TeX

G:=Group("C2^4.413C2^3");

// GroupNames label

G:=SmallGroup(128,1446);

// by ID

G=gap.SmallGroup(128,1446);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,268,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=g^2=a,f^2=b*a=a*b,a*c=c*a,e*d*e=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=a*b*d,f*g=g*f>;

// generators/relations

G = C24.413C23 order 128 = 27

253rd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₁₃C₂³ order 128 = 2⁷

253^rd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²